Consider sampling every hyper-edge of \(\mathcal {H}\)  with probability \((C/\gamma )n^{1-k}\)  for a constant \(C\)  to be specified later. Let the sampled hypergraph be \(\mathcal {L}\) . First note that \(\max _{v_1,v_2}\mathrm {codeg}_{\mathcal {L}}(v_1,v_2)\)  is stochastically dominated by \(\operatorname{Bin}(n^{k-2},(C/\gamma )n^{-(k-1)})\)  or \(\operatorname{Bin}(n^{k-2},n^{-k+5/4})\)  for \(n\)  sufficiently large in terms of \(C\) . This implies that as there are \(n^2\)  pairs to union bound over, that with high probability all codegrees in \(\mathcal {L}\)  are bounded by 3.
We now consider the subset of edges in \(\mathcal {L}\)  where every vertex has \(\mathcal {L}\) -degree at most \(C+C^{3/4}\) ; call this hypergraph \(\mathcal {L}^{\prime }\) . Furthermore by Chernoff and Markov with probability at least \(7/8\)  we have that \((1-e^{-\Omega (C^{1/3})})n\)  vertices have \(\mathcal {L}\) -degree in \([C\pm C^{3/4}]\) . Now we wish to show that taking the induced (sampled) hypergraph \(\mathcal {L}^{\prime }\)  does not dramatically reduce the degree of the vast majority of these vertices.
Reveal a vertex \(v\)  and assume that \(\max _{v^{\prime }\ne v}\mathrm {codeg}(v,v^{\prime })\le 3\)  and its degree is in \([C\pm C^{3/4}]\) . Having revealed the outcome of edges coming out of \(v\) , we need to prove that it is highly likely that the remaining neighbors in all these edges have degree at most \(C+C^{3/4}\) , so that the \(\mathcal {L}^{\prime }\) -degree of \(v\)  is the same as the \(\mathcal {L}\) -degree and hence is in \([C\pm C^{3/4}]\) . This occurs for \(v\)  with probability \(1-e^{-\Omega (C^{1/3})}\)  by Chernoff, having revealed every edge containing \(v\)  (using that the original degrees in \(\mathcal {H}\)  are tightly controlled).
Therefore we find from Markov that with probability \(3/4\) , say, \(\mathcal {L}^{\prime }\)  has at least \((1-e^{-\Omega (C^{1/3})})n\)  vertices with degree within \(C\pm C^{3/4}\) , no vertices with degree above \(C+C^{3/4}\) , and all codegrees bounded by 3. The desired result then follows via applying [1]} to the non-isolated vertices of \(\mathcal {L}^{\prime }\) , which gives a sparse vertex cover, and then deleting any hyperedge in the cover containing a vertex which is covered more than once. We obtain a partial hypergraph matching which is evidently spread due to the sampling of hyperedges defining \(\mathcal {L}\) .
